The Development of Intuitionistic Logic

We will be principally concerned with the historical
development of the intuitionists’ explanation of the logical
connectives. An “explanation” here is an account of what one knows
when one understands and correctly uses the logical connectives. The
emphasis is on (the history of) Brouwer's explanation of logic
within the framework of intuitionistic mathematics, and on (the
history of) its codification in Heyting's Proof Interpretation.

In future updates of this entry, the following topics will be added: (a)
precursors to Brouwer, (b) early objections to the Proof
Interpretation, and (c) later developments around the Proof
Interpretation.

The standard explanation of intuitionistic logic today is the
BHK-Interpretation (for “Brouwer, Heyting, Kolmogorov”) or Proof
Interpretation as given by Troelstra and Van Dalen in
Constructivism in mathematics (Troelstra and van Dalen 1988, 9):

(H1)

A proof of A ∧ B is given by presenting a proof of A and a
proof of B.

(H2)

A proof of A ∨ B is given by presenting either a proof of A
or a proof of B (plus the stipulation that we want to regard the
proof presented as evidence for A ∨ B).

(H3)

A proof of A → B is a construction which
permits us to transform any proof of A into a proof of
B.

(H4)

Absurdity ⊥ (contradiction) has no proof; a proof of
¬A is a construction which transforms any hypothetical
proof of A into a proof of a contradiction.

(H5)

A proof of ∀xA(x) is a
construction which transforms a proof of d ∈ D
(D the intended range of the variable x) into a
proof of A(d).

(H6)

A proof of ∃xA(x) is given by
providing d ∈ D, and a proof of
A(d).

Notions such as “construction”, “presenting”
and “transformation” can be understood in different ways,
and indeed they have been. Similarly, there have been different ideas
as to how one may justify that concrete instances of clauses H3 and H4
indeed work for any (possibly hypothetical) proof of the
antecedent. Logical principles that are valid on one understanding of
these notions may not be valid on another. As Troelstra and Van Dalen
indicate, it is even possible to understand these clauses in such a
way that they validate the principles of classical logic (Troelstra
and van Dalen 1988, 9,
32–33).[1]
In the context of the foundational programs
of intuitionism and constructivism, all notions are of course
understood to be effective; but even then there is room for
differences of understanding. Such differences can have mathematical
consequences. On some understandings, intuitionistic logic turns out
formally to be a subsystem of classical logic (namely, classical logic
without the Principle of the Excluded Middle). But that is not the
understanding of intuitionistic mathematicians, who, in analysis, have
constructed intuitionistically valid instances of the schema
¬∀x(Px ∨
¬Px), while classically there can be none (see
the section on
Strong counterexamples and the Creating Subject,
below).

Troelstra and Van Dalen specify that the clauses H1–H6 go back to
Heyting's explanation from 1934 (hence “H”). Heyting's aim had been
to clarify the conception of logic in Brouwer's foundational program
in mathematics, which would motivate adding the following clause:

(H0)

A proof of an atomic proposition A is given by presenting a
mathematical construction in Brouwer's sense that makes A true.

Indeed, as we will see, a version of the Proof Interpretation
is implicit already in Brouwer's early writings from 1907 and 1908,
and was notably used by him in his proofs of the bar theorem from
1924 and 1927, which predate Heyting's papers on logic. We will
therefore begin our account of the historical development of
intuitionistic logic with Brouwer's ideas, and then show how, via
Heyting and others, the modern Proof Interpretation was arrived at.

As Sundholm (1983, 159) points out,
in the terms “BHK-Interpretation” and “Proof Interpretation” it
would be appropriate to replace “Interpretation” by “Explanation”.
For in a logical-mathematical context, “interpretation” has come to
refer to the interpretation of one formal theory in
another.[2]
An interpretation of a formal system U in a formal
system V is given by a translation ′ of formulas
of U to formulas of V that preserves
provability:[3]

If U
⊢
A then V
⊢
A′

For the moment, we note that the BHK-Interpretation or Proof
Interpretation is not an interpretation in this mathematical sense,
but is rather a meaning explanation; we will come back to such
interpretations and their difference from explanations in
section 4.5.2 below.

While accepting Sundholm's point, we keep the terms themselves,
considering that they have perhaps become too common to change.
Section 5.3
below is the appropriate place to
explain our preference for “Proof Interpretation” over
“BHK-Interpretation”.

The name “Proof Interpretation” for the explanation that Heyting
published in the 1930s and later seems to have made its first
appearance in print only in 1973, in papers by Van Dalen and Kleene,
presented at the same conference (van Dalen 1973a, Kleene 1973).
Heyting himself spoke simply of the “interpretation”
(1958A, 107; 1974, 87) or “the intuitionistic interpretation”
(1958A, 110) of logic.

The name “BHK-Interpretation” was coined by Troelstra
(1977a, 977), where “K” initially stood for
“Kreisel” (because of Kreisel 1962a), later for
“Kolmogorov”, e.g., in Troelstra 1990 (p.6); in a future
update of this entry, it will be explained that this replacement is,
in keeping with Sundholm's point, a correction.

In his dissertation Brouwer 1907,
Brouwer presents his conception of the relations between
mathematics, language, and logic. Both the intuitionistic view of
logic as essentially sterile, and the existence of results in
intuitionistic logic that are incompatible with classical logic,
depend essentially on that conception.

For Brouwer, pure mathematics consists primarily in the act of
making certain mental constructions (Brouwer 1907, 99n.1)/(Brouwer
1975, 61,
n.1).[4]
The point of departure for these constructions is the intuition of
the flow of
time.[5]
This intuition, when divested from all sensuous content, allows us to
perceive the form “one thing and again a thing, and a continuum
in between”. Brouwer calls this form, which unites the discrete
and the continuous, “the empty two-ity”. It is the basic
intuition of mathematics; the discrete cannot be reduced to the
continuous, nor the continuous to the discrete (Brouwer
1907, 8)/(Brouwer 1975, 17).

As time flows on, an empty two-ity can be taken as one part of a
new two-ity, and so on. The development of intuitionistic
mathematics consists in the exploration which specific constructions
the empty two-ity and its self-unfolding or iteration allows and
which not:

The only possible foundation of mathematics must be
sought in this construction under the obligation carefully to watch
which constructions intuition allows and which not. (Brouwer
1907, 77)/(Brouwer 1975, 52)

or, in Heyting's words,

[Brouwer's] construction of intuitionist mathematics is
nothing more nor less than an investigation of the utmost limits
which the intellect can attain in its self-unfolding. (Heyting
1968A, 314)

Brouwer and other intuitionists have shown how on this basis
arithmetic, real analysis, and topology can be constructed.
Moreover, Brouwer considers any exact thought that is not itself
mathematics an application of mathematics. For whenever we
consciously think of two things together while keeping them
separate, we do so, according to Brouwer, by projecting the empty
two-ity on them (Brouwer 1907, 179n.1)/(Brouwer 1975, 97n.1).

Brouwer takes the intuition of time to belong to pre-linguistic
consciousness. Mathematics, therefore, is essentially languageless.
It is the activity of effecting non-linguistic constructions out of
something that is not of a linguistic nature. Using language we can
describe our mathematical activities, but these activities
themselves do not depend on linguistic elements, and nothing that is
true about mathematical constructional activities owes its truth to
some linguistic fact. Linguistic objects such as axioms may serve to
describe a mental construction, but they cannot bring it into being.
For this reason, certain axioms from classical mathematics are
rejected by intuitionists, such as the completeness axiom for real
numbers, which says that if a non-empty set of real numbers has an
upper bound, then it has a least upper bound: we know of no general
method that would allow us to construct mentally the least upper
bound whose existence the axiom claims.

As Brouwer later put it, “Formal language accompanies
mathematics as the weather-map accompanies the atmospheric
processes” (Brouwer 1975, 451). Correspondingly, establishing
properties of formal systems may have many uses, but ultimately has
no foundational significance for mathematics. In a lecture from
1923, Brouwer expresses optimism about Hilbert's proof theory, but
denies that it would have significance for mathematics:

We need by no means despair of
reaching this goal [of a consistency proof for formalized
mathematics], but nothing of mathematical value will thus be gained:
an incorrect theory, even if it cannot be inhibited by any
contradiction that would refute it, is none the less incorrect, just
as a criminal policy is none the less criminal even if it cannot be
inhibited by any court that would curb it. (Brouwer 1924N, 3)/(van
Heijenoort 1967, 336)

At the same time, Brouwer was well aware of the practical need for
language, both in order to communicate mathematical results to
others and to help ourselves in remembering and reconstructing our
previous results (Brouwer 1907, 169)/(Brouwer 1975, 92).
Only an ideal mathematician with perfect and unlimited memory would
be able to practice pure mathematics without recourse to language
(Brouwer 1933A2, 58)/(Brouwer 1975, 443). Clearly, given
these two practical functions of language, the more precise the
language is, the better.

Logic, in this framework, seeks and systematizes certain patterns
in the linguistic recordings of our activities of mathematical
construction. It is an application of mathematics to the language of
mathematics. Specifically, logic studies the patterns that
characterize valid inference. The aim is to establish general rules
operating on statements about mathematical constructions such that, if
the original statements (the premises) convey a mathematical truth, so
will the statement obtained by applying the rule (the conclusion)
(Brouwer 1949C,1243). What is preserved in an inference from given
premises to a conclusion is therefore not, as in classical logic, a
kind of possibly evidence-transcendent truth, but
constructibility. This view is quite explicit already in Brouwer's
dissertation (Brouwer 1907, 125–132, 159–160)/(Brouwer 1975,
72–75, 88), but a more memorable passage is in the paper from 1908:

Is it allowed, in purely mathematical constructions and
transformations, to neglect for some time the idea of the
mathematical system under construction and to operate in the
corresponding linguistic structure, following the principles of
syllogism, of contradiction and of tertium exclusum, and can we then
have confidence that each part of the argument can be justified by
recalling to the mind the corresponding mathematical construction?
(Brouwer 1908, 4)/(Brouwer 1975, 109)

(He then goes on to argue that the answer is “yes” for
the principles of the syllogism and of contradiction, but, in general,
“no” for the Princple of the Excluded Middle (PEM); more
on this below,
section 2.4.)

But if a certain mathematical construction can be constructed
out of another one, this is a purely mathematical fact, and as such
independent of logic. Logic therefore is descriptive but not
creative: by the use of logic, one will never obtain mathematical
truths that are not obtainable by a direct mathematical construction
(Brouwer 1949C, 1243). Hence, in the development of intuitionistic
mathematics, logic can never play an essential role. It follows from
Brouwer's view that logic is subordinate to mathematics. The
classical view that mathematics is subordinate to logic is closely
related to the view that pure logic has no particular subject matter
or domain, and is prior to all. From that perspective, Brouwer's
conception of logic as dependent on mathematics will seem too
restrictive. But for Brouwer logic always presupposes mathematics,
because in his view it is, like any exact thought, an application of
mathematics.

The resulting linguistic system of logic may in turn be studied
mathematically, even independently of the mathematical activities
and their recordings that it was originally abstracted from.
Iterating the process, an infinite hierarchy arises of mathematical
activities, their linguistic recordings, and the mathematical study
of these recordings as linguistic objects independently of their
original meaning. Brouwer describes this hierarchy (in more detail
than we have done here) at the end of his dissertation (Brouwer
1907, 173ff), and criticizes Hilbert for not respecting it. Of
particular interest is the distinction Brouwer makes between
mathematics and “mathematics of the second order” (Brouwer
1907, 99n.1, 173)/(Brouwer 1975, 61n.1, 94), where the latter is the
mathematical study of the language of the former in abstraction from
its original meaning; this way, Brouwer made fully explicit the
distinction between mathematics and (what became known as)
metamathematics (e.g., Hilbert 1923 (p.153). Later, Brouwer claimed
priority for this distinction, adding in a footnote that he had
explained it to Hilbert in a series of conversations in 1909
(Brouwer 1928A2, 375)/(Mancosu 1998, 44n.1).

Brouwer realized
(Brouwer 1907, 125–128)/(Brouwer 1975, 72–73) that
the hypothetical judgement seems to pose a problem for his view on
logic as described above. For what is peculiar to the hypothetical
judgement, Brouwer says, is that there the priority of mathematics
over logic seems to be reversed. Among the examples he refers to are
the proofs found in elementary geometry of the problems of
Apollonius. Here is one of them: Given three circles, defined by
their centers and their radii, construct a fourth circle that is
tangent to each of the given three. The way this is usually solved
is first to assume that such a fourth circle exists, then to set up
equations that express how it is related to the three given circles,
and then, via algebraic manipulations and logic, arrive at explicit
definitions of the center and radius of the required circle, and,
from there, at corresponding mathematical constructions. So it seems
that here one first has to assume the existence of the required
circle, then use logic to make various judgements about it, and only
thereby arrives at a mathematical construction for it.

However, Brouwer argues, this is not what really happens. His
general interpretation of such cases is as follows. Having first
remarked that logical reasoning accompanies or mirrors mathematical
activity which is at least conceptually prior to that reasoning,
Brouwer then says:

There is a special case [… ] which really seems
to presuppose the hypothetical judgment from logic. This occurs
where a structure in a structure is defined by some relation,
without it being immediately clear how to effect its construction.
Here one seems to assume to have effected the required
construction, and to deduce from this hypothesis a chain of
hypothetical judgments. But this is no more than apparent; what one
is really doing in this case is the following: one starts by
constructing a system that fulfills part of the required relations,
and tries to deduce from these relations, by means of tautologies,
other relations, in such a way that in the end the deduced
relations, combined with those that have not yet been used, yield a
system of conditions, suitable as a starting-point for the
construction of the required system. Only by this construction will
it then have been proved that the original conditions can indeed be
satisfied. (Brouwer 1907, 126–127)/(Brouwer 1975, 72 (modified))

Different readings of this concise passage have been proposed.
According to one, Brouwer's passage bears on A →
B in the following way:

(α) Brouwer points out in the above
lines that if the conditions and specifications for A are
given, then we try to add more information in such a way that, after
a certain amount of constructional activity, we can really carry out
a construction of A which respects the specifications. Once
this is accomplished, we can turn to the “implication” construction
for B, which yields the construction for B and to
the required embedding of the structure for A into the
structure for B. (van Dalen 2004, 250–251)

According to interpretation α, A → B
just means A ∧ B with the extra information that
the construction for B was obtained from that for
A. On this reading A → B can be
asserted only after a construction for A has been found. The
idea is clear: namely, to avoid hypothetical constructions, and the
use of logic they require, by insisting that a construction be
supplied that proves the antecedent. (As we will see in a future
update of this entry, Freudenthal (1936) too has suggested this
strategy, albeit with a different motivation.) But, as Van Dalen also
notices, it is also in effect a rejection of the hypothetical
judgement in the general case where one does not know whether there is
a construction for A.

An alternative reading is β:

(β) In order to establish A →
B, one has to conceive of A and B as
conditions on constructions, and to show that from the conditions
specified by A one obtains the conditions specified by
B, according to transformations whose composition preserves
mathematical constructibility. (van Atten in press)

On this reading, Brouwer's explanation of the hypothetical judgement
avoids hypothetical constructions and the concomitant use of logic
by considering conditions on constructions instead of constructions
themselves. Instead of a “chain of hypothetical judgements” that one
seems to make, one is really making a chain of transformations in
which from required relations (i.e., given conditions) further
relations are derived.

On either reading, it is clear that Brouwer had the Proof
Interpretation of the implication in mind already in 1907; for on
either reading, the essential component in the explanation
of A → B is the preservation of
constructibility from A to B. For further discussion
of Brouwer's passage on the hypothetical judgement and the two
readings of it mentioned here, see Kuiper 2004, van Dalen 2004, van
Atten in press, and van Dalen 2008.

Intuitionistically, to say that a proposition A is true
is primarily to say that we have effected a construction that is
correctly described by A; the proposition A is made
true by the construction. Idealizing to a certain extent, we say that
A is true if we possess a construction method that, when
effected, will yield a construction that is correctly described by
A. According to Brouwer, to say that a proposition A
is false then must mean that it is impossible to effect an appropriate
construction; notation ¬A. Such an impossibility is
recognized either immediately (e.g., the impossibility to identify 1
unit and 2 units) or mediately. In the former case, one observes
directly that an intended construction is blocked; it “does not
go through” (Brouwer 1907, 127)/(Brouwer 1975, 73). In the
latter case, one shows that a proposition A is contradictory
by reducing A to a known falsehood, e.g., one shows that
A → 1=2 (Brouwer 1954A, 3). In practice, one defines
¬A := A → 1=2 (and hence ¬1=2 is seen as a
particular case of A → A).

The notion of “negation as impossibility” is known as
“strong negation”. One speaks of the “weak
negation” of A to express that so far no proof
of A has been found. This excludes neither finding a proof
of A nor finding a proof of ¬A later. Clearly,
then, to assert the weak negation of A is not to assign a
truth value besides true and false to it; Barzin and Errera's claim
(see
section 4.3
below) that its treatment of negation turns Brouwer's logic into a
three-valued one is groundless. The distinction between weak and
strong negation is important for the so-called “weak
counterexamples”.

As the rules of logic operate on
linguistic objects, and these linguistic objects may be considered
separately from the precise mathematical context in which they
described a truth, it is possible to apply the rules of logic and
obtain new linguistic objects without providing a precise
mathematical context for the latter. In other words, the logical
principles, which can be stated without specifying the context in
which they are applied, and thereby suggest context-independence,
are for their correctness sensitive to the context. There is no
general guarantee that logical principles which are valid in one
context, will be equally valid in a different one. This is what
Brouwer means when he speaks of “the unreliability of the logical
principles”, the title and theme of his seminal paper
Brouwer 1908; see also Brouwer 1949A (p. 1243).

In that paper, Brouwer draws a consequence of his general view
on logic that he had overlooked in his dissertation: PEM, A ∨
¬A, is not valid. Its constructive validity would mean
that we have a method that, for any A, either gives us a
construction for A, or shows that such a construction is
impossible. But we do not have such a general decision method, and
there are many open problems in mathematics. Brouwer states “Every
number is finite or infinite” as an example of a general proposition
for which so far no constructive proof has been found. As a
consequence, he says, it is at present uncertain whether problems
such as the following are solvable:

Is there in the decimal expansion of π a
digit which occurs more often than any other one?

Do there occur in the decimal expansion of π infinitely many
pairs of consecutive equal digits?
(Brouwer 1908, 7)/(Brouwer 1975, 110)

In effect, Brouwer is saying that we can assert the weak negations
of the propositions expressed in these questions; hence, these
propositions are so-called “Brouwerian counterexamples” or
“weak counterexamples” to PEM. On the constructive reading
of PEM, of course any as yet unsolved problem is a weak counterexample
to PEM. Brouwer began to publish weak counterexamples to PEM in
international journals only much later (Brouwer 1921A, Brouwer 1924N,
Brouwer 1925E).

Brouwer remarks in the 1908 paper that the fact that PEM is not
valid does not mean that it is false: ¬(A ∨ ¬
A) implies ¬A ∧ ¬¬A, a
contradiction. In other words, ¬¬(A ∨ ¬A)
is correct. Brouwer concludes that it is always consistent to use
PEM but that it does not always lead to truths. In the latter case,
the argument that appeals to PEM establishes not the truth, but the
consistency of its conclusion. Brouwer proposes to divide the
theorems that are usually considered as proved into the true and the
non-contradictory ones (Brouwer 1908, 7n.2)/(Brouwer 1975, 110n.2).
That is not a suggestion that there are three truth values, true,
non-contradictory, false; for a non-contradictory proposition might
be proved one day and thereby become true.

A mathematical context in which PEM is valid, Brouwer points
out, is that of the question whether a given construction of finite
character is possible in a given finite domain. In such a context
there are only finitely many possible attempts at that construction,
and each will succeed or fail in finitely many steps (for clarity,
the phrasing here is not that of Brouwer 1908 but that of Brouwer
1955). So on these grounds A ∨ ¬A holds, where
A is the proposition stating that the construction exists.

Brouwer ascribed the belief in the general validity of PEM to an
unwarranted projection from such finite cases (in particular, those
arising from the application of finite mathematics to everyday
phenomena) to the
infinite.[6]

In his dissertation of 1907, Brouwer still accepted PEM as a
tautology, understanding A ∨ ¬A as ¬A
→ ¬A (Brouwer 1907, 131, 160)/(Brouwer
1975, 75,
88).[7]
Curiously, he did realize at the same time that there is no evidence
for the principle that every mathematical proposition is either
provable or refutable (Brouwer 1907, 142n.3)/(Brouwer 1975, 101); this
principle is the constructively correct reading of PEM. In the
paper from 1908, he corrected his earlier understanding of PEM:

Now consider the principium tertii exculsi: it claims
that every supposition is either true or false; in mathematics this
means that for every supposed embedding of a system into another,
satisfying certain conditions, we can either accomplish such an
embedding by a construction, or we can arrive by a construction at the
arrestment of the process which would lead to the embedding. (Brouwer
1908, 5)/(Brouwer 1975, 109)

It follows
that the question of the validity of the principium tertii
exclusi is equivalent to the question whether unsolvable
mathematical problems can exist. There is not a shred of
proof for the conviction, which has sometimes been put
forward [here Brouwer
refers in a footnote to Hilbert 1900b] that there exist no unsolvable mathematical
problems.

Here he seems to overlook that, constructively,
there is a difference between the claim that every mathematical
problem is solvable and the weaker claim that there are no absolutely
unsolvable problems. The former is equivalent to A ∨
¬A, the latter to ¬¬(A ∨
¬A); and Brouwer had demonstrated the intuitionistic
validity of the latter in the same paper. Indeed, in the Brouwer
archive there is a note from about the same period 1907–1908 in
which the point is made explicitly:

Can one ever demonstrate of a proposition, that it can
never be decided? No, because one would have to so by reductio ad
absurdum. So one would have to say: assume that the proposition has
been decided in sense a, and from that deduce a
contradiction. But then it would have been proved that
not-a is true, and the proposition is decided after all.
(van Dalen 2001, 174n.a) [translation mine]

Brouwer never published this note. Wavre in 1926 gave the argument
for a particular case, clearly seeing the general point:

It suffices to give an example of a number of which one
does not know whether it is algebraic or transcendent in order to
give at the same time an example of a number that, until further
information comes in, could be neither the one nor the other. But,
on the other hand, it would be in vain, it seems to me, to want to
define a number that indeed is neither algebraic nor transcendent,
as the only way to show that it is not algebraic consists in showing
that it is absurd that it would be, and then the number would be
transcendent. (Wavre 1926, 66) [translation mine]

The explicit observation that ¬¬(A ∨
¬A) means that no absolutely unsolvable problem can be
indicated was made in Heyting 1934 (p. 16).

Three examples can be given that show that by the mid-1920s, Brouwer
in practice worked with the hypothetical judgement and with the
clause for implication in the Proof Interpretation (which was
published later): an equivalence in propositional logic, the proof
of the bar theorem, and his reading of ordering axioms.

In a lecture in 1923, Brouwer presented
a proof of ¬¬¬A ⇔ ¬A (Brouwer
1925E, 253)/(Mancosu 1998,
291).[8]
This equivalence is the one theorem in propositional logic that
Brouwer ever published. The argument begins by pointing out
that A → B implies that ¬B →
¬A (because ¬B is B → ⊥
and → is transitive). It would not have been possible
for Brouwer to make this inference if at the time it would have been
among his proof conditions of an implication to have a proof of the
antecedent, as then a proof of A → B would lead
to a proof of B and thereby make it impossible to begin
establishing the second implication by proving its antecedent
¬B.

Later, Brouwer pointed out the following consequence of the validity
of ¬¬¬A ⇔ ¬A: the proof method of
reductio ad absurdum, which is not generally correct (not
“reliable”) intuitionistically, can be used to establish negative
propositions ¬A
(Brouwer 1929A, 163)/(Mancosu 1998, 52). For if the
assumption of ¬¬A leads to a contradiction, that is, to
¬¬¬A, the equivalence allows one to simplify that to
¬A.

Brouwer's bar theorem is crucial to intuitionistic analysis; for a
detailed explanation of the notions involved and of Brouwer's proof,
see Heyting 1956 (Ch. 3), Parsons 1967, and van Atten 2004b
(Ch.4). Here we will rather be concerned with the logical aspects.

Brouwer's proof of the bar theorem from 1924 (later versions of the
proof appeared in 1927 and in 1954) proves a statement of the form
“If A has been demonstrated, then B is
demonstrable” (Brouwer 1924D1, Brouwer 1927B, Brouwer
1954A). This is evidently not the same as an implication A
→ B understood as a transformation of the proof
conditions of A into those of B, because in the
former case there is the additional information that, by
hypothesis, A has been demonstrated. In other words, we have,
by hypothesis, a concrete proof of A at hand. (However, both
are hypothetical judgements in the sense that neither requires that we
actually have demonstrated A.) It may be possible to exploit
this extra information, and below it will be indicated how Brouwer did
this. (Heyting in 1956 chose to define implication in this stronger
sense; see
section 5.4 below.)

The statement in question can be rendered more specifically as
“If it has been demonstrated that tree T contains a set of
nodes forming a bar, then it can be demonstrated that T
contains a bar that is
well-ordered”.[9]
Brouwer first formulates a condition for any demonstration that may be
found of the proposition “Tree T contains a
bar”. This condition is that any demonstration of that
proposition must be analyzable into a certain canonical form. Brouwer
then gives a method to transform any such demonstration, when analyzed
into that canonical form, into a mathematical construction that makes
the proposition “T contains a well-ordered bar”,
true, thereby establishing the consequent. This strategy clearly shows
that Brouwer's operative explanation of the meaning of A
→ B was a version of clause (H3) of the Proof
Interpretation as formulated in the Introduction, if we understand
“proof” in that clause as “demonstration”.

A demonstration or concrete proof of the antecedent, be it an
actual or a hypothetical one, is required to obtain a canonical
form. The reason is that the existence of a canonical proof of a
proposition A cannot be logically derived from the mere
proof conditions of A, as the form such a canonical proof
takes may well depend on specific non-logical details of
mathematical constructions for A.

In Brouwer's proof of the bar theorem, the applicability of the
transformation method to any demonstration of the antecedent is
guaranteed by the fact that the condition on such demonstrations
that he formulates is a necessary condition. Brouwer obtained this
necessary condition by exploiting the fact that on his conception,
mathematical objects, so in particular trees and bars, are mental
objects; this opens the possibility that reflection on the way these
objects and their properties are constructed in mental acts provides
information on them that can be put to mathematical use, in
particular if this information consists in constraints on these acts
of construction. This is how Brouwer arrived at his canonical form.
In effect, Brouwer's argument for the bar theorem is a
transcendental argument. On other conceptions of mathematics such
considerations need not be acceptable, and indeed no proofs of the
(classically valid) bar theorem are known in other varieties of
constructive mathematics (where bar induction is either accepted as
an axiom, a possibility that Brouwer had also suggested (Brouwer
1927B, 63n.7)/(van Heijenoort 1967, 460n.7), or not accepted, as in the Markov School).

For a more detailed discussion of this matter, see Sundholm and
van Atten 2008.

In a lecture course on Order Types in 1925, of which David van
Dantzig's notes are preserved in the Brouwer Archive, Brouwer
commented:

The axioms II through V are to be understood in the constructive
sense: if the premisses of the axiom are satisfied, the virtually
ordered set should provide a construction for the order condition in
the conclusion. (van Dalen 2008, 19)

This is a clear instance of the clause for implication in the Proof
Interpretation. Note that Brouwer did not include this elucidation
in the published paper (1926A), nor in later
presentations.

As we saw above, in the paper from 1908 Brouwer had given weak
counterexamples to PEM. In the 1920s Brouwer developed a general
technique for constructing weak counterexamples which also made it
possible to widen their scope and include principles of analysis. The
development began in 1921, when Brouwer gave a weak counterexample to
the proposition that every real number has a decimal expansion
(Brouwer 1921A). The argument proceeded by defining real numbers whose
decimal developments are dependent on specific open problems
concerning the decimal development of π. Brouwer closed by
observing that, should these open problems be solved, then other real
numbers without decimal expansion can be defined (Brouwer 1921A,
210)/(Mancosu 1998, 34). The general technique was made explicit in a
lecture from 1923 (Brouwer 1924N,3 and footnote 4)/(van Heijenoort
1967,337 and footnote 5) and reached its perfection with the
“oscillatory number” method in the first Vienna lecture in
1928 (Brouwer 1929A, 161)/(Mancosu 1998, 51). The method involves the
reduction of the validity of a mathematical principle to the
solvability of an open problem of the following type: we have a
decidable property P (defined on the natural numbers) for
which we have as yet shown neither
∃xP(x) nor
∀x¬P(x). This reduction is
carried out in such a way that it only uses the fact that P
induces an open problem of this type, and does not depend on the exact
definition of P; that is, if the open problem is solved, one
can simply replace it by another one (of the same type), and exactly
the same reduction still works. This uniformity means that, as long as
there are open problems of this type at all (and this is practically
certain at any time), there is no intuitionistic proof of the general
mathematical principle in question. In the 1920s, Brouwer constructed
weak counterexamples to the following general mathematical
propositions, among others (where R stands for the
set of intuitionistic real numbers, and Q for the set
of rationals):

The continuum is totally ordered (Brouwer 1924N)

Every set is either finite or infinite (Brouwer 1924N)

The Heine-Borel theorem (Brouwer 1924N)

∀x ∈ R(x
∈ Q ∨ x ∉ Q)
(Brouwer 1925E)

Any two straight lines in the Euclidian plane are either parallel,
or coincide, or intersect (Brouwer 1929A)

Every infinite sequence of positive numbers either converges
or diverges (Brouwer 1929A)

A weak counterexample shows that we cannot at present prove some
proposition, but it does not actually refute it; in that sense, it
is not a counterexample proper. From 1928 on, Brouwer devised a
number of strong counterexamples to classically valid propositions,
that is, he showed that these propositions were contradictory. This
should be understood as follows: if one keeps to the letter of the
classical principle but in its interpretation substitutes
intuitionistic notions for their classical counterparts, one arrives
at a contradiction. So Brouwer's strong counterexamples are no more
counterexamples in the strict sense of the word than his weak
counterexamples are (but for a different reason). One way of looking
at strong counterexamples is that they are non-interpretability
results.

That strong counterexamples to classical principles are possible at
all is explained as follows. As mentioned, on the intuitionistic
understanding, logic is subordinate to mathematics, whereas
classically it is the other way around. Hence, if intuitionistic
mathematics contains objects and principles that do not figure in
classical mathematics, it may come about that intuitionistic logic,
which then depends also on these non-classical elements, is no
proper part of classical logic.

Brouwer's first strong counterexample was published in Brouwer
1928A2, where he showed:

¬∀x∈R(x
∈ Q ∨ x ∉
Q)

This is a strengthening of the corresponding weak counterexample from
1923, but the argument is entirely different. The strong
counterexample depends on the theorem in intuitionistic analysis,
obtained in 1924 and improved in 1927, that all total functions [0,1]
→ R are uniformly continuous. The non-classical
elements in that theorem are the conception of the continuum as a
spread of choice sequences, and the Bar Theorem based on it (for
further explanations of this conception, see Heyting 1956 (Ch.3) and
van Atten 2004b (Chs.3 and
4)).[10]

From 1948 on, Brouwer also published counterexamples that are
based on so-called “creating subject methods”. (He
mentions in Brouwer 1948A that he has been using this method in
lectures since 1927.) Their characteristic property is that they make
explicit reference to the subject who carries out the mathematical
constructions, to the temporal structure of its activities, and the
relation of this structure to the intuitionistic notion of truth.
These methods can be used to generate weak as well as strong
counterexamples. (In the earlier “oscillatory number”
method for generating weak counterexamples, the creating subject is
not explicitly referred to.)

Using creating subject methods, Brouwer showed, for instance,

¬∀x∈R(¬¬ x
> 0 → x > 0) (Brouwer 1949A)

¬∀x∈R(x ≠ 0
→ x < 0 ∨ x > 0) (Brouwer 1949B)

As the actual arguments using these methods quickly get somewhat
complicated, but introduce no new logical phenomena as such —
weak and strong counterexamples can also be given by other means
— we refer to the literature for further details: Brouwer
1949A, Brouwer 1949B, Brouwer 1954F; Heyting 1956 (pp. 117–120);
Myhill 1968; Dummett 2000a (pp. 244–245). We do note here one
particular aspect of this method. It seems to introduce a further
notion of negation, by accepting that, if it is known that the
creating subject will never prove A, then A is
false. But this is actually no different from the notion of negation
as impossibility. Heuristically, this can be seen as follows: given
the freedom the creating subject has to construct whatever it can, the
only way to show that there can be no moment at which the subject
demonstrates A must be to show that a demonstration of
A itself is impossible. An actual justification of the
principle is this: If the creating subject demonstrates a proposition
A, it does so at a particular moment n; so, by
contraposition, if it is contradictory that there exists a moment
n at which the subject demonstrates A, then
A is
contradictory.[11]

In Brouwer 1955, the four possible cases a proposition
α may be in at any particular moment are made explicit:

α has been proved to be true;

α has been proved to be false, i.e. absurd;

α has neither been proved to be true nor to be absurd, but
an algorithm is known leading to a decision either that α is
true or that α is absurd;

α has neither been proved to be true nor to be absurd, nor
do we know an algorithm leading to the statement either that
α is true or that α is absurd.

In a lecture from 1951, Brouwer lists only cases 1, 2, and 4 from the
above list, adding that case 3 “obviously is reducible to the
first and second cases” (Brouwer 1981A, 92). That remark
emphasizes an important idealization permitted in intuitionistic
mathematics: we may make the idealization that, once we have obtained
a decision method for a specific proposition, we also know its
outcome.

Brouwer also adds that a proposition for which case 4 holds, may at
some point pass to another case, either because we have in the
meantime found a decision method, or because the objects involved in
proposition α in the meantime have acquired further
properties that permit to make the decision (as may happen for
propositions about choice sequences).

In 1908, Brouwer had shown that ¬¬(A ∨
¬ A); in 1923, when Hilbert's program was in full swing,
this result led Brouwer to say that “We need by no means despair
of reaching this goal [of a consistency proof for formalized
mathematics]”; see section 2.1 above for
the
full quotation.
(At the time, Brouwer
suspected that ¬¬A → A was weaker than
PEM; Bernays quickly corrected this impression in a letter to Brouwer
(Brouwer 1925E, 252n.4)/(Mancosu 1998, 292n.4).)

In 1928, he added to this the consistency of finite conjunctions
of instances of PEM, and considered these results to “offer some
encouragement” for the formalist project of a consistency proof
(Brouwer 1928A2, 377)/(Mancosu 1998,
43).[12]
The strongest statement based on these
results he made at the end of the first Vienna lecture from 1928:

An appropriate mechanization of the language of this
intuitionistically non-contradictory mathematics should therefore
deliver precisely what the formalist school has set as its goal.
(Brouwer 1929A, 164) [translation mine]

But, for reasons explained above, such a consistency proof would
have no mathematical value for Brouwer; and the best a classical
mathematician can be said to be doing, according to the view Brouwer
sketches, is to be giving relative consistency proofs.

Gödel's incompleteness theorems showed that Hilbert's
Program, in its most ambitious form, cannot succeed. Brouwer's
assistant Hurewicz discussed the incompleteness theorem in a seminar
(van Dalen 2005, 674n.7). There is no comment from Brouwer on
Gödel's first theorem in print; on the other hand, he clearly
had the second theorem in mind when he wrote, in 1952, that

The hope originally fostered by the Old Formalists that
mathematical science erected according to their principles would be
crowned one day with a proof of non-contradictority, was never
fulfilled, and, nowadays, in view of the results of certain
investigations of the last few decades, has, I think, been
relinquished. (Brouwer 1952B, 508)

Hao Wang reports:

In the spring of 1961 I visited Brouwer at his home. He
discoursed widely on many subjects. Among other things he said that
he did not think G's incompleteness results are as important as
Heyting's formalization of intuitionistic reasoning, because to him
G's results are obvious (obviously true). (Wang 1987,
88)[13]

With respect to the first incompleteness theorem, Brouwer's
reaction is readily understandable. Already in his dissertation, he
had noted that the totality of all possible mathematical constructions
is “denumerably unfinished”; by this he meant that
“we can never construct in a well-defined way more than a
denumerable subset of it, but when we have constructed such a subset,
we can immediately deduce from it, following some previously defined
mathematical process, new elements which are counted to the original
set” (Brouwer 1907, 148)/(Brouwer 1975, 82). And in one of the
notebooks leading up to his dissertation, he stated that “The
totality of mathematical theorems is, among other things, also a set
which is denumerable but never
finished”.[14]

Indeed, according to Carnap, it had been an argument of Brouwer's
that had stimulated Gödel in finding the first theorem. In a
diary note for December 12, 1929, Carnap states that Gödel talked
to him that day “about the inexhaustibility of mathematics (see
separate sheet) He was stimulated to this idea by Brouwer's Vienna
lecture. Mathematics is not completely formalizable. He appears to be
right” (Wang 1987, 84). On the “separate sheet”,
Carnap wrote down what Gödel had told him:

We admit as legitimate mathematics certain reflections on
the grammar of a language that concerns the empirical. If one seeks to
formalize such a mathematics, then with each formalization there are
problems, which one can understand and express in ordinary language,
but cannot express in the given formalized language. It follows
(Brouwer) that mathematics is inexhaustible: one must always again
draw afresh from the “fountain of intuition”. There is,
therefore, no characteristica universalis for the whole
mathematics, and no decision procedure for the whole mathematics. In
each and every closed language there are only countably many
expressions. The continuum appears only in “the whole
of mathematics” … If we have only one language,
and can only make “elucidations” about it, then these
elucidations are inexhaustible, they always require some new intuition
again. [As quoted, in translation, in Wang 1987 (p. 50)]

This record contains in particular elements from the second of
Brouwer's two lectures in Vienna, in which one finds the argument
that Gödel refers to: on the one hand, the full continuum is given
in a priori intuition, while on the other hand, it cannot be
exhausted by a language with countably many expressions
(Brouwer 1930A, 3, 6)/(Mancosu 1998, 56, 58).

The second incompleteness theorem, on the other hand, must have
surprised Brouwer, given his optimism in the 1920s about the formalist
school achieving its aim of proving the consistency of formalized
classical mathematics (see the quotation at the beginning of this
subsection).

In his final original published paper (1955), Brouwer was, in
his own way, quite positive about the study of classical logic.
After showing that various principles from the algebraic tradition
in logic (e.g., Boole, Schröder) are intuitionistically
contradictory, he continues:

Fortunately classical algebra of logic has its merits
quite apart from the question of its applicability to mathematics.
Not only as a formal image of the technique of common-sensical
thinking has it reached a high degree of perfection, but also in
itself, as an edifice of thought, it is a thing of exceptional
harmony and beauty. Indeed, its successor, the sumptuous symbolic
logic of the twentieth century which at present is continually
raising the most captivating problems and making the most surprising
and penetrating discoveries, likewise is for a great part cultivated
for its own sake. (Brouwer 1955, 116)

Brouwer's logic has played a role in the Grundlagenstreit (the
Foundational Debate) only to the extent that this logic could be
seen as a fragment of classical logic. Constructive logic in that
sense was a success, and it became fundamental to Hilbert's Program
as well. On the other hand, phenomena specific to Brouwer's full
conception of logic, in particular the strong counterexamples,
played no role in the Foundational Debate whatsoever. The main
reason for this may be that, in their dependence on choice
sequences, they use objects that are not acceptable in classical
mathematics. (A more subtle matter is whether they are acceptable in
Hilbert's finitary mathematics. According to Bernays, Hilbert never
took a position on choice sequences (Gödel 2003, 279), and
more generally never read Brouwer's papers
(van Dalen 2005,
637).[15])
In addition, Brouwer did not announce the existence of strong
counterexamples in a loud or polemical way; and when in 1954 he
finally did publish (in English) a paper with a polemical title
— “An example of contradictority in classical theory of
functions” — , the Foundational Debate was, in the social
sense, long over. Intuitionistic logic and mathematics had been
widely accepted to the extent that they could be seen as the
constructive part of classical mathematics, while the typically
intuitionistic innovations were ignored. It is not surprising, then,
that the presentation of the strong counterexamples in the 1950s did
not at all lead to a reopening of the debate. For further discussion
of this matter, see Hesseling 2003 and van Atten 2004a.

As Brouwer was more interested in developing pure mathematics than
logic, which for him was a form of applied mathematics, he never
made an extensive study of the latter, and in particular never made
a systematic comparison between on the one hand intuitionistic logic
and on the other hand classical logic as formalized, for example, in
Principia Mathematica (Whitehead and Russell 1910) or by the
Hilbert school (Hilbert 1923,Hilbert and Ackermann 1928). What
motivated others to make such comparisons was the publication by
Brouwer in international journals of weak counterexamples that
showed how these also affected very general mathematical principles
such as trichotomy for real numbers (see above,
section 3.2).

Clearly, to make a systematic comparison possible, one needs a
codification of intuitionistic logic in a formal system. On the
intuitionistic view that cannot exist, as logic is as open-ended as
the mathematics it depends on. But one may formalize fragments of
intuitionistic logic. The relevant papers here are Kolmogorov 1925,
Heyting 1928 (unpublished), Glivenko 1928, Glivenko 1929, and
Heyting
1930.[16]
But perhaps the first to give systematic thought to the matter was
Paul Bernays. In a letter to Heyting of November 5, 1930, he wrote:

The lectures that Prof. Brouwer at the time held in
Göttingen (for the first time) [1924 (van Dalen 2001, 305)], led
me to the question how a Brouwerian propositional logic could be
separated out, and I arrived at the result that this can be done by
leaving out the single formula ¬¬a ⊃ a (in
your symbolism). I then also wrote to Prof. Brouwer [correcting
Brouwer's impression at the time that this formula is weaker than
PEM]. (Troelstra 1990,
8)[17]

(Bernays’ correction was included, at the proof stage, in Brouwer's
paper (Brouwer 1925E, 252n.4)/(Mancosu 1998, 292n.4).)
However, Bernays did not publish his idea for a Brouwerian logic.
(Kolmogorov would publish the same idea in 1925; see below.)

In setting up a formal system to capture, albeit necessarily
only in part, logic as it figures in Brouwer's foundations,
naturally some priorly obtained meaning explanation is needed to
serve as the criterion for intuitionistic validity. Yet none of the
papers by Kolmogorov, Heyting, and Glivenko just mentioned made an
explicit contribution to the meaning explanation of intuitionistic
logic. As we will see, the explanations as given in the papers
(which is not necessarily all their respective authors had in mind)
were too vague for that. It is perhaps not surprising, then, that
the systems were not equivalent; notably, Kolmogorov rejected Ex
Falso, while Heyting and Glivenko accepted it. We will now discuss
these papers in turn.

In 1925, Andrei Kolmogorov, at the age of 22, published the first
(partial) formalization of intuitionistic logic, and also made an
extensive comparison with formalized classical logic, in a paper
called “On the principle of the excluded middle”. As Van
Dalen has suggested (Hesseling 2003, 237), Kolmogorov probably had
come into contact with intuitionism through Alexandrov or Urysohn, who
were close friends of Brouwer's. Kolmogorov was in any case remarkably
well-informed, citing even papers that had only appeared in the
Dutch-language “Verhandelingen” of the Dutch Royal Academy
of Sciences (Brouwer 1918B, Brouwer 1919A, Brouwer 1921A).

The task Kolmogorov set himself in the paper is to explain why “the
illegitimate use of the principle of the excluded middle” as
revealed in Brouwer's writings “has not yet led to contradictions
and also why the very illegitimacy has often gone unnoticed”
(van Heijenoort 1967, 416). In effect, as other passages make
clear (van Heijenoort 1967, 429–430), the (unachieved) aim is to
show that classical mathematics is translatable into intuitionistic
mathematics, and thereby give a consistency proof of classical
mathematics relative to intuitionistic mathematics.

The technical result established in the paper is: Classical
propositional logic is interpretable in an intuitionistically
acceptable fragment of it. The intuitionistic fragment,
called B (presumably for “Brouwer”)
is:

A → (B → A)

(A → (A → B)) →
(A → B)

(A → (B → C)) →
(B → (A → C))

(B → C) → ((A
→ B) → (A → C))

(A → B) → ((A →
¬B) → ¬A)

The system H (presumably for
“Hilbert”) consists of B and the
additional axiom

6. ¬¬A → A

In both systems, the rules are modus ponens and substitution.

Kolmogorov then indicates and partially carries out a proof that
H is equivalent to the system for classical
propositional logic presented by Hilbert in Hilbert 1923.
Then Kolmogorov devises the following translation *:

A*

=

¬¬A for atomic A

F(φ1,
φ2,… ,φk)*

=

¬¬F(φ1*,
φ2*,
… ,φk*) for composed
formulas

and proves this interpretability result:

If U ⊢H
φ then
U* ⊢B
φ*

where U is a set of axioms of H, and
U* the set of its translations (which Kolmogorov
shows to be derivable in B).

Kolmogorov's technical result anticipated Gödel's and Gentzen's
“double negation translations” for arithmetic (see below), all the
more so since he also made quite concrete suggestions how to treat
predicate logic. As Hesseling (2003, 239) points out,
however, to see Kolmogorov's result as a translation into
intuitionistic mathematics is slightly different from his own way of
seeing it. Kolmogorov saw it as a translation into a domain of
“pseudomathematics”; but although he did not explicitly identify
that as part of intuitionistic mathematics, he could have done so.

Kolmogorov's strategy to obtain a (fragment) of formalized
intuitionistic logic was to start with a classical system and isolate
an intuitionistically acceptable system from it. (Note that, although
Kolmogorov refers to Principia Mathematica, he did not take
it as his point of departure.) This might (roughly) be described as
the method of crossing out, which is also what Heyting would do in
1928 (see below). Given the task Kolmogorov set himself, it is a
natural approach. Kolmogorov's criterion whether to keep an axiom was
whether a proposition has an “intuitive foundation” or
“possesses intuitive obviousness” (van Heijenoort 1967,
421, 422); on implication he said, “The meaning of the
symbol A → B is exhausted by the fact that,
once convinced of the truth of
A, we have to accept the truth of B too” (van
Heijenoort 1967,420). No more precise indications are given, so in
that sense the paper did little to explain the meaning of
intuitionistic logic.

Ex Falso was excluded from this fragment: Kolmogorov said that,
just like PEM, Ex Falso “has no intuitive foundation” (van
Heijenoort 1967, 419). In particular, he says that Ex Falso is
unacceptable for the reason that it asserts something about the
consequences of something impossible (van Heijenoort 1967, 421).
Note that that is a very strong rejection: it not only rules out Ex
Falso in its full generality, but also specific instances such as
“If 3.15 is an initial segment of π, then 3.1 is an
initial segment of π”. It also indicates
an incoherence in Kolmogorov's position: one cannot at the same time accept
A → (B → A) as an axiom and deny
that anything can be asserted about the consequences of an
impossible B.

It is not known whether Kolmogorov sent his paper to Brouwer
(van Dalen 2005, 555). The contents of the paper seem to have
remained largely unknown outside the Soviet Union for years.
Glivenko mentioned the paper in a letter to Heyting of October 13,
1928, as did Kolmogorov in an undated letter to Heyting of 1933 or
later (Troelstra 1990, 16); but in Heyting 1934 it is, unlike
Kolmogorov's later paper from 1932, neither discussed nor included
in the bibliography. The volume of the Jahrbuch über die
Fortschritte der Mathematik covering 1925, which included a
very short notice on Kolmogorov 1925 by V. Smirnov (Leningrad),
wasn’t actually published until 1932 (Smirnov 1925).

While Kolmogorov's work remained unknown in the West, an independent
initiative towards the formalization of intuitionistic logic and
mathematics was taken in 1927, when the Dutch Mathematical Society
chose to pose the following problem for its annual contest:

By its very nature, Brouwer's set theory cannot be
identified with the conclusions formally derivable in a certain
pasigraphic system [i.e., universal notation system]. Nevertheless
certain regularities may be observed in the language which Brouwer
uses to give expression to his mathematical intuition; these
regularities may be codified in a formal mathematical system. It is
asked to

construct such a system and to indicate its deviations
from Brouwer's theories;

to investigate whether from the system to be constructed a
dual system may be obtained by (formally) interchanging the
principium tertii exclusi and the principium
contradictionis. (Troelstra 1990, 4)

The question had been formulated by Brouwer's friend, colleague, and
former teacher Gerrit Mannoury, who asked Brouwer's opinion on it
beforehand in a letter (Brouwer was in
Berlin);[18]
unfortunately, no reply from Brouwer has been found, but the final
formulation was the same as in Mannoury's letter.

Brouwer's former student Arend Heyting, who had graduated (cum
laude) in 1925 with a dissertation on intuitionistic projective
geometry, wrote what turned out to be the one submission (Hesseling
2003, 274). The original manuscript seems no longer to exist, but it
is known that its telling motto was “Stones for
bread”.[19]
In 1928, the jury crowned Heyting's
work,[20]
stating that it was “a formalization
carried out in a most knowledgeable way and with admirable
perseverance” (Hesseling 2003, 274) [translation modified].

Heyting's essay covered propositional logic, predicate logic,
arithmetic, and Brouwerian set theory or analysis. One would think
that, to be able to achieve this, Heyting must have had quite
precise ideas on how to explain the logical connectives
intuitionistically. However, Heyting's correspondence with
Freudenthal in 1930
shows that before 1930, Heyting had not yet arrived at the explicit
requirement of a transformation procedure in the explanation of
implication (see the
quotation in
section 5.1 below).

Since the original manuscript seems not to have survived, a
discussion of Heyting's work must take the revised and published
version from 1930 as its point of departure; see below.

Heyting sent his manuscript to Brouwer, who replied in a letter
of July 17, 1928, that he had found it “extraordinarily
interesting”, and continued:

By now I have already begun to appreciate your work so
much, that I should like to request that you revise it in German for
the Mathematische Annalen (preferably somewhat extended rather than
shortened).[21]

Interestingly, Brouwer also suggested (with an eye on the
formalization of the theory of choice sequences)

And, with an eye on §13, perhaps also the notion of
“law” can be formalized.

It seems, however, that Heyting made no effort in that direction.

Heyting's paper would indeed be published soon after, in 1930;
not in the Mathematische Annalen, as Brouwer by then was no longer
on its editorial board, but in the proceedings of the Prussian
Academy of Sciences. However, Heyting's work became known already
before its publication. Heyting mentioned it in correspondence with
Glivenko in 1928 (see below), Tarski and Łukasiewicz talked about
it to Bernays at the Bologna conference in 1928, and Church
mentioned it in a letter to Errera in 1929 (Hesseling 2003, 274).

In reaction to Barzin and Errera, who had
argued that Brouwer's logic was three-valued and moreover that this
led to it being inconsistent, Valerii
Glivenko[22]
in 1928 set out to prove them wrong by formal means. He gave the
following list of axioms for intuitionistic propositional logic:

p → p

(p → q) → ((q
→ r) → (p → r))

(p ∧ q) → p

(p ∧ q) → q

(r → p) → ((r
→ q) → (r → (p
∧ q)))

p → (p ∨ q)

q → (p ∨ q)

(p → r) →
((q → r) → ((p ∨ q)
→ r))

(p → q) → ((p →
¬ q) → ¬ p)

From these axioms, he then proved

¬¬(p ∨ ¬p)

¬¬¬p → ¬p

((¬p ∨ p) → ¬q) →
¬q

The first two had already informally been argued for by Brouwer
(Brouwer 1908, Brouwer 1925E); the third was new. Now suppose that in
intuitionistic logic, a proposition may be true (p holds),
false (¬p holds), or have a third truth value
(p′ holds). Clearly, p →
¬p′ and ¬p →
¬p′, and therefore (¬p
∨ p) → ¬p′; but then, by the third
of the lemmata above and
axiom 8
in the list, ¬p′. As p is arbitrary, this means
no proposition has the third truth value. (In 1932, Gödel would
show that, more generally, intuitionistic propositional logic is
no n-valued logic for any natural number
n; see
section 4.5.1 below.)

Like Kolmogorov in 1925 and, as we will see, Heyting in 1930,
Glivenko provided no detailed explanation of the intuitionistic
validity of these axioms.

Glivenko immediately continued this line of work with a second
short paper, Glivenko 1929, in which he showed, for a richer system
of intuitionistic propositional logic:

If p is provable in classical propositional logic, then
¬¬p is provable in intuitionistic propositional logic;

If ¬p is provable in classical propositional logic, then
it is also provable in intuitionistic propositional logic.

The first theorem is not a translation in the usual sense (as
Kolmogorov's is), as it does not translate subformulas of p; but
it is strong enough to show that the classical and intuitionistic
systems in question are equiconsistent.

The system of intuitionistic propositional logic is richer than in
Glivenko's previous paper, because to its axioms have now been added
the following four:

(p → (q → r)) →
(q → (p → r))

(p → (p → r)) →
(p → r)

p → (q → p)

¬q → (q → p)

Interestingly, Glivenko mentions in his paper that “It is
Mr. A. Heyting who first made me see the appropriateness of the two
axioms C and D in the Brouwerian logic” (Mancosu
1998, 304–305n.3). The two had come into correspondence when,
upon the publication of Glivenko 1928, Heyting sent Glivenko a letter
(Troelstra 1990, 11). Kolmogorov in 1925 had explicitly rejected Ex
Falso for having no “intuitive foundation”. From
Glivenko's letter to Heyting of October 13, 1928, we know that
Glivenko was aware of this (Troelstra 1990, 12). In his paper, however,
which he finished later, he does not mention Kolmogorov at
all. Instead, he makes the remark on Heyting just quoted and then
justifies D by saying that it is a consequence of the principle
(p ∨ ¬q) → (q
→ p), the admissibility of which he considers
“quite
evident”.[23]
From a Brouwerian point of view, however, the principle is as
objectionable as Ex Falso.[24]
It is worth noting that, when Heyting gave his
justification for Ex Falso in Heyting 1956 (p. 102), he did
not appeal to the principle Glivenko had used (nor did Kolmogorov in 1932). From
Glivenko's letter of October 18, 1928, one gets the impression that
this principle had not been the argument Heyting had actually given
to convince him:

I am now convinced by your reasons that intuitionistic mathematics
need not reject that axiom, so that all considerations against that
axiom might lead beyond the limits of our present subject matter.
(Troelstra 1990, 12) [translation mine]

Heyting had informed Glivenko of the planned publication of his
(revised and translated) prize essay from 1928 in the Mathematische
Annalen. On October 30, 1928, Glivenko asked Heyting if he also was
going to include the result that if p is provable in classical
propositional logic, then ¬¬ p is provable in intuitionistic
propositional logic; for if he would, then there would be no point
for Glivenko in publishing his own manuscript. Two weeks later,
Glivenko had changed his mind, writing to Heyting on November 13
that, even though this result “is but an almost trivial remark”,
“its rigorous demonstration is a bit long” and he wants to publish
it independently of Heyting's paper. Indeed, Glivenko's paper was
published first, and in it the publication of Heyting's
formalization was announced; and when Heyting published his paper in
1930, he included a reference to Glivenko 1929, stating its two
theorems, and he also acknowledged the use of results from Glivenko
1928. Heyting's note “On intuitionistic logic”, also from 1930,
begins with a reference to Glivenko's “two excellent articles” from
1928 and 1929.

Heyting's (partial) formalization of intuitionistic logic and
mathematics in Heyting 1930, Heyting 1930A, and
Heyting 1930B, is perhaps, as far as the parts on logic are
concerned, the most influential intuitionistic publication ever,
together with his book Intuitionism. An introduction from
1956.

Heyting's formalization comprised intuitionistic propositional
and predicate logic, arithmetic, and analysis, all together in one
big system (with only variables for arbitrary objects). The part
concerned with analysis was, not only in its intended interpretation
(involving choice sequences) but also formally, no subsystem of its
classical counterpart; this explains why it sparked no general
interest at the time. (A consequence we noted above is that
Brouwer's strong counterexamples never affected the debate.)
Therefore this part of Heyting's formalization was left out of
consideration by the other participants in the Foundational
Debate.[25]
This was different for the parts concerned with logic and
arithmetic. Formally speaking and disregarding their intended
interpretations, from these one could distill subsystems of their
classical counterparts, from which only PEM (or double negation
elimination) is missing. No doubt this encouraged many to accord to
these systems a definitive character, which, as Heyting had remarked
at the beginning of his paper, on the intuitionistic conception of
logic they cannot have:

Intuitionistic mathematics is a mental activity
[“Denktätigkeit”], and for it every language,
including the formalistic one, is only a tool for communication. It is
in principle impossible to set up a system of formulas that would be
equivalent to intuitionistic mathematics, for the possibilities of
thought cannot be reduced to a finite number of rules set up in
advance. Because of this, the attempt to reproduce the most important
parts of formal language is justified exclusively by the greater
conciseness and determinateness of the latter vis-à-vis
ordinary language; and these are properties that facilitate the
penetration into the intuitionistic concepts and the use of these
concepts in research. (Heyting 1930, 42)/(Mancosu 1998, 311)

However, Heyting himself, however, wrote some five decades later,

I regret that my name is known to-day mainly in
connection with these papers [(Heyting 1930), (Heyting 1930A),
(Heyting 1930B)], which were very imperfect and contained many
mistakes. They were of little help in the struggle to which I
devoted my life, namely a better understanding and appreciation of
Brouwer's ideas. They diverted the attention from the underlying
ideas to the formal system itself. (Heyting 1978, 15)

The fear that the attention might be thus diverted had indeed been
expressed in the first of the three papers themselves:

Section 4 [on negation] departs substantially from
classical logic. Here I could not avoid giving the impression that
the differences that come to the fore in this section constitute the
most important point of conflict between intuitionists and
formalists (a claim that is already refuted by the remark made at
the beginning [quoted above]); this impression arises because the
formalism is unfit to express the more fundamental points of
conflict. (Heyting 1930, 44)/(Mancosu 1998, 313)

For the full system, including predicate logic and analysis, the
reader is referred to Heyting's original papers. Heyting's axioms
for intuitionistic propositional logic were:

a → (a ∧ a)

(a ∧ b) → (b
∧ a)

(a → b) → ((a
∧ c)→ (b ∧ c))

((a → b) ∧ (b
→ c)) → (a → c)

b → (a
→ b)

(a ∧ (a → b))
→ b

a → (a ∨ b)

(a ∨ b) → (b ∨ a)

((a → c) ∧ (b
→ c)) → ((a ∨ b)
→ c)

¬a → (a
→ b)

((a → b) ∧ (a →
¬b)) → ¬a

In a letter to Oskar Becker, Heyting described the approach used to
obtain these axioms, as well as those for predicate logic, as
follows:

I sifted the axioms and theorems of Principia
Mathematica and, on the basis of those that were found to be
admissible, looked for a system of independent axioms. Given the
relative completeness of Principia, this to my mind ensures the
completeness of my system in the best possible way. Indeed, as a
matter of principle, it is impossible to be certain that one has
captured all admissible modes of inference in one formal
system. [Heyting to Becker, September 23, 1933 (draft) (van Atten
2005, 129), Original italics, translation mine]

As Heyting emphasizes here, the theorems of Principia
Mathematica also had to be looked at, for a theorem might be
intuitionistically acceptable even when a classical proof given for
it is
not.[26]
It is worth noting
that Heyting used this method of crossing out, as also Kolmogorov
had, instead of determining the logic directly from general
considerations on mathematical constructions in Brouwer's sense. (To
some extent, Kreisel tried to do that systematically in the 1960s;
this will be discussed in a future update of this entry.) In his
dissertation on intuitionistic axiomatization of projective
geometry, Heyting had already gained experience with developing an
intuitionistic system by taking a classical axiomatic system as
guideline and then adjusting
it.[27]

In Mints 2006 (section 2) it has been observed that Russell 1903
(section 18) anticipated intuitionistic propositional logic by
identifying Peirce's law and using it to separate out the principles
that imply PEM. It seems that Heyting did not realize this at the
time; among the references given in Heyting 1930, Russell's book
does not appear.

Heyting shows the independence of his axioms using a method
given by Bernays (1926); this use of a non-intended interpretation
for metamathematical purposes Heyting accepted without hesitation,
but he remarked that such metamathematics is “less important for us
[intuitionists]” than the approach where all formulas are considered
to be meaningful propositions (Heyting 1930, 43)/(Mancosu 1998, 312).

Heyting states Glivenko's two theorems from 1929, without giving
proofs.

Unlike Kolmogorov, but like Glivenko (who had been convinced by
Heyting), Heyting accepted Ex Falso
(axiom 10 above).
He was somewhat more elaborate on this point than they had
been:

The formula a → b generally means:
“If a is correct, then b is correct
too.” This proposition is meaningful if a and
b are constant propositions about whose correctness nothing
need be known … The case if conceivable that after the
statement a → b has been proved in the sense
specified, it turns out that b is always correct. One
accepted, the formula a → b then has to remain
correct; that is, we must attribute a meaning to the sign → such
that a → b still holds. The same can be
remarked in the case where it later turns out that a is
always false. For these reasons, the formulas
[5] and
[10] are
accepted. (Heyting 1930, 44)/(Mancosu 1998, 313)

The argument is, however, incomplete. It is uncontroversial that,
once a → b has been proved, it should remain so
when afterward ¬ a is proved. But why should a
→ b, if it has not yet been proved, become provable just
by establishing ¬a? (Johansson asked this in a letter to
Heyting of September 23,
1935;[28]
the matter will be
discussed in a future update of this entry.) Clearly, then, further
work needed to be done on the explanation of intuitionistic logic.

After the publication of Heyting's series of papers, three roads
could be taken, and indeed were (cf. Posy 1992):

to explicate and develop the meaning
explanation that had given rise to Heyting's system;

to engage in metamathematical study of the formal systems distilled from
Heyting's system;

to find alternative motivations for (parts of) Heyting's system
that are independent from the intuitionists’, yet also in some
sense constructive (e.g., Lorenzen's dialogue semantics)

By and large, these three roads lead to very different areas, with
correspondingly divergent histories, of which no unified account can
be expected. (However, in the Dialectica Interpretation, as proposed
and understood by Gödel (1958, 1970, 1972), they came close to
one another; this will be treated in a future update of this entry.)
In accordance with the main theme of the present account, in the
remaining sections we will be concerned with the historical
development of the meaning explanations. But a number of early
highlights of the formal turn must be mentioned here.

Intuitionistic propositional logic is not a finitely valued logic.
Gödel (1932) showed that Heyting's system for intuitionistic
propositional logic cannot be conceived of as a finitely many-valued
logic. Apparently unbeknown to Gödel, this confirmed a conjecture
of Oskar Becker (1927, 775–777). Gödel's result, which
came soon after his incompleteness theorem, led Heyting to write to
him, “It is as if you had a malicious pleasure in showing the
purposelessness of others’ investigations … In the sense
of economy of thought this work is certainly useful, and in addition
to that comes the particular beauty of your short proof”
[Heyting to Gödel, letter dated 24 and 26 November 1932 (Gödel
2003a, 67).

Peano Arithmetic is translatable into Heyting Arithmetic. A
seminal theorem was obtained independently by Gödel (1933e) and
by Gentzen (withdrawn upon learning of Gödel's paper): There is a
translation ′ from PA to HA
such that

PA ⊢
A ⇔ HA
⊢ A′

(Gödel and Gentzen actually used Herbrand's axioms for the natural
numbers (Herbrand 1931), but that is immaterial here.) The same
proofs serve to show that the same result holds for pure predicate
logic. This completes and generalizes Kolmogorov's result from 1925,
which at the time was known to neither Gentzen nor Gödel. (Gödel
does cite Glivenko 1929.)

Gödel concluded that “the system of intuitionistic
arithmetic and number theory is only apparently narrower than the
classical one, and in truth contains it, albeit with a somewhat
deviant interpretation” (Gödel 1933e, 37)/(Gödel 1986,
295). Heyting replied that “for the intuitionist, the
interpretation is what is essential” (Heyting 1934,18,
trl. mine). Later Gödel became, as Kreisel put it,
“supersensitive about differences in meaning” (Kreisel
1987a,82).
The Gödel-Gentzen translation had an immediate application to the
foundational debate, in which besides the notion of existence the
status of PEM had been the main issue (Hesseling 2003): the embedding
of PA into HA shows
that PA is consistent if and only if
HA is. As Gödel observed,

The above considerations, of course,
provide an intuitionistic consistency proof for classical arithmetic
and number theory. This proof, however, is not “finitary” in the
sense in which Herbrand, following Hilbert, used the term.
(Gödel 1933e, 37–38)/(Gödel 1986,
295)[29]

Indeed, according to Bernays (1967, 502), it was this fact that made
it clear that intuitionism is stronger than Hilbert's finitism.

For more details on this result, see the section
Gödel's Work in Intuitionistic Logic and Arithmetic of
Kennedy 2007, as well as Artemov 2001.

As pointed out by Troelstra (Gödel 1986, 299), at the time
Gödel certainly knew the content of Heyting 1931. He had
attended the Königsberg conference (where Heyting had presented
that paper) and had published a review (Gödel 1932f) of the
printed version. In that paper, Heyting had introduced a provability
operator, but considered it redundant given the intuitionistic
conception of truth as provability
(see below,
section 5.1).[30]

Consider the following two properties:

the Disjunction Property (DP): if
⊢
A ∨ B, then
⊢
A or
⊢ B;

the Explicit Definability property (EP): if
⊢ ∃xP(x), then
⊢
P(t) for some term t

DP for intuitionistic propositional calculus was stated by Gödel
(1932, 66)/(1986, 225), who did not give a proof. A proof was given for
intuitionistic predicate logic by Gentzen (1934). As Troelstra and Van
Dalen (1988, 175) observe, the same method (cut-elimination) yields ED
for intuitionistic predicate calculus. For Heyting Arithmetic, see
Kleene 1945 (section 8) and Kleene 1952 (theorem 62b).

The intuitionistic connectives are not interdefinable: none of
→, ∧, ∨, ¬ can be defined in terms of the
others. Heyting had stated this in Heyting 1930 (p. 44)
(Mancosu 1998, 312), but without giving a proof. A proof was
published by Wajsberg (1938) and (independently and
by different methods) by McKinsey (1939).

Various mathematical interpretations (in the sense explained in
section 1.2)
of formalized intuitionistic logic and arithmetic have been
proposed. Above we saw Gödel's translation of intuitionistic
propositional logic into the classical modal logic S4
from 1933; further examples are Kleene's realizability (Kleene 1945),
and Gödel's Dialectica Interpretation (Gödel 1958, Gödel
1970, Gödel 1972) (which will be discussed in a future update of this
entry).

Such mathematical interpretations are not meaning explanations.
There are two arguments. The first is that, in a context where formal
systems are arithmetized, interpretability results are established
wholly within mathematics (e.g., Joosten 2004). But then no contact at
all is made with the concepts that figure in meaning explanations,
which have to do with our cognitive capacities, such as those of
effecting mathematical constructions or those of understanding,
learning, and using a language. The second, more general argument is
due to Sundholm (1983, 159). Here we consider A and its
interpretation A′ not as syntactical objects but as
meaningful propositions. Then the argument is that, by whatever means
one correlates a mathematical proposition A′ to a
mathematical proposition A, it makes sense to ask whether the
propositions A and A′ are equivalent. On the
other hand, it does not make sense to ask whether the
proposition A is equivalent to a specification of what one
has to know in order to understand A and use it correctly;
for example, that specification remains the same whether A is true,
false, or undecided. But then A′ cannot be a meaning
explanation of A.

It would therefore be a mistake to see in mathematical
interpretations of intuitionistic systems ways to make the Proof
Interpretation “rigorous” or “precise”. The
difference is not one of degree but of
kind.[31]
The point is general, and it makes no
difference if the interpreting theory V is
intuitionistic.[32]
On the other hand, it is not at all ruled out that for an interpreting
theory V itself an explanation can be given that is arguably
superior to the Proof Interpretation in some respect; this was
Gödel's philosophical aim for his Dialectica Interpretation
(which will be treated in a future update of this entry).

A related point can be made about model-theoretic semantics for
a(ny) logic (Dummett 1973, 293–294): these map formulas to
mathematical objects, without there being an intrinsic connection
between those objects and the concepts related to our understanding
and use of a language. By itself, therefore, such a semantics does not
contribute to a meaning explanation. (The first argument on
mathematical interpretations, above, is essentially the same as this
one.) But for metamathematical purposes, such model-theoretic
semantics have proved extremely valuable. Note that Heyting (1930)
used model-theoretic semantics to show the independence of his
axioms for propositional logic. A wide variety of model-theoretic
semantics for intuitionistic systems has been developed since,
beginning with the topological ones of Stone (1937) and Tarski
(1938). Of the remaining ones, among the best known are Beth models
(Beth 1956), Kripke models (Kripke 1965), and topos models. For Kripke
models, see the section Kripke semantics for intuitionistic
logic of Moschovakis 2007; for other models and further
references, see Kleene and Vesley 1965 (p. 6) and Artemov 2001.

Heyting told Van Dalen that he had the notion of (intuitionistic)
construction in mind to guide him in devising his formalization of
intuitionistic logic and mathematics in 1927. In the published version
of his formalization, he did not elaborate much on the meaning of the
connectives; all he explained there about the general meaning
of a → b was that “If a is
correct, then b is correct too” (Heyting
1930, 44)/(Mancosu 1998, 313). Correspondence between Heyting and
Freudenthal in 1930 shows that Heyting up till then did not have a
more refined explanation at hand; we will come back to this later in
this section.

Heyting began to expound on the meaning of the connectives in print
in two papers written in 1930. The first, Heyting 1930C,
was published that same year, the second, the text of his lecture at
Königsberg in September 1930, was published the next year
(Heyting 1931).

The first paper was a reaction to Barzin and Errera's claim that
Brouwer's logic is three-valued (Barzin and Errera 1927). The
relevant points for the explanation of the meaning of the
connectives in Heyting's paper are the following. First, an
explanation is given of assertion:

Here, then, is the Brouwerian assertion of
p: It is known how to prove p. We will
denote this by
⊢p.
The words “to prove” must be taken in the sense of
“to prove by construction”. [original italics](Heyting
1930C, 959)/(Mancosu 1998, 307)

And then of intuitionistic negation:

⊢¬p
will mean: “It is known how to reduce p to a
contradiction.” (Heyting 1930C, 960)/(Mancosu
1998, 307)

Heyting goes on to explain that, although on these explanations
there is a third case beside
⊢p and
⊢
¬p, namely
the case where one knows neither how to prove p nor how to refute
it, this does not mean there is a third truth value:

This case could be denoted by p′, but it
must be realized that p′ will hardly ever be a
definitive statement, since it is necessary to take into account the
possibility that the proof of either p or ¬p
might one day succeed. If one does not wish to risk having to
retract what one has said, in the case p′ one should
not state anything at all. (Heyting 1930C, 960)/(Mancosu 1998, 307)

This refutes the contention of Barzin and Errera. Note that these
points are all in Brouwer's writings, too. Indeed, Heyting (1932, 121)
objects to Barzin and Errera's term “Heyting's logic”,
saying that “all the fundamental ideas of that logic come from
Brouwer” (translation mine). But Heyting's papers will have
found a wider audience than Brouwer's. Brouwer, in turn, was very
positive about the paper Heyting 1930C, and wrote to the editor of the
journal in which it appeared:

While preparing a note on intuitionism for the Bulletin
de l’Académie Royale de
Belgique,[33]
I was pleasantly surprised to see the publication of a note by my
student Mr. Heyting, which elucidates in a magisterial manner the
points that I wanted to shed light upon myself. I believe that after
Heyting's note little remains to be said. [Brouwer to De Donder,
October 9, 1930, trl. van Dalen 2005 (p. 676)]

Heyting also proposes a provability operator +, where +p
means “p is provable”. The distinction
between p and +p is relevant if one believes that
(at least some) propositions are true or false independently of our
mathematical activity. In that case one can go on and develop a
provability logic, as for example Gödel did
(see
section 4.5.1 above).
That is not the intuitionistic conception, and Heyting remarks that,
if the fulfilment of p requires a construction, then there is
no difference between p and +p. He adds that, on the
intuitionistic explanation of negation, there is indeed no difference
between ¬p and +¬p, as a proof of
¬p is defined as a construction that reduces p
to a contradiction. But Heyting does not generalize this remark to all
of intuitionistic logic. The final section of the paper is a further
discussion of the logic of the provability operator, in particular its
interaction with negation (e.g.,
⊢
¬+p is the assertion that p is unprovable). But
Heyting ends by saying that, as the intuitionists’ task is the
reconstruction of all mathematics, while at the same time no examples
of propositions have been found so far for which this provability
operator would be necessary to express their status (e.g., to express
absolute undecidability), it cannot be asked of intuitionists that
they develop this logic (Heyting 1930C, 963)/(Mancosu
1998, 309–310)

The Königsberg lecture, given in 1930 and published in 1931,
specifies the meanings of p, ¬p, and p
∨ q. This time Heyting makes an explicit connection to
phenomenology:

We here distinguish between propositions [Aussagen] and
assertions [Sätze]. An assertion is the affirmation of a
proposition. A mathematical proposition expresses a certain
expectation. For example, the proposition, “Euler's constant
C is rational”, expresses the expectation that we could
find two integers a and b such that
C = a / b. Perhaps the word
“intention”, coined by the phenomenologists, expresses
even better what is meant here … The affirmation of a
proposition means the fulfillment of an intention. (Heyting
1931, 113)/(Benacerraf and Putnam 1983, 58–59)

Compared to the earlier paper written in 1930, the point about the
provability operator is amplified:

The distinction between p and +p
vanishes as soon a construction is intended in p itself,
for the possibility of a construction can be proved only by its
actual execution. If we limit ourselves to those propositions which
require a construction, the logical function of provability does not
arise at all. We can impose this restriction by treating only
propositions of the form “p is provable”, or, to put it
another way, by regarding every intention as having the intention of
a construction for its fulfilment added to it. It is in this sense
that intuitionistic logic, insofar as it has been developed up to
now without using the function +, must be understood.
(Heyting 1931, 115)/(Benacerraf and Putnam 1983, 60) (modified)

The explanation of disjunction in the Königsberg lecture is:

“p ∨ q” signifies that
intention which is fulfilled if and only if at least one of the
intentions p and q is fulfilled. (Heyting
1931, 114)/(Benacerraf and Putnam 1983, 59)

And of negation:

Becker, following Husserl, has described its meaning very
clearly. For him negation is something thoroughly positive, viz., the
intention of a contradiction contained in the original intention. The
proposition “C is not rational”, therefore,
signifies the expectation that one can derive a contradiction from the
assumption [Annahme] that C is rational. It is important to
note that the negation of a proposition always refers to a proof
procedure which leads to the contradiction, even if the original
proposition mentions no proof procedure. (Heyting 1931,
113)/(Benacerraf and Putnam 1983, 59)

Heyting pointed out that these explanations for disjunction and
negation, taken together, are an immediate argument against the
acceptability of PEM, for which a general method would be needed that,
applied to any given proposition p, produces either a proof
of p or a proof of ¬p. What Heyting did not do
here was to generalize this explanation of negation to one for
implication. Also, note that the procedure does not operate on proofs
of p, but starts from merely the assumption that p,
which in general gives less information. Both points were taken care
of shortly afterward. In a letter to Freudenthal dated October 25,
1930, shortly after the Königsberg lecture, Heyting wrote:

From your remarks it has
become clear to me that the simple explanation of
a → b by “When I think a, I must
think b” is untenable; this idea is in any case too
indeterminate to be able to serve as the foundation for a logic. But
also your formulation: “When a has been proved, so
has b”, is not wholly satisfactory to me; when I ask
myself what you may mean by that, I believe that also a
→ b, like the negation, should refer to a proof
procedure: “I possess a construction that derives from every
proof of a a proof of b”. In the following, I
will keep to this interpretation. There is therefore no difference
between a → b and +a →
+b. (Troelstra 1983a, 206–207) [translation mine]

This explanation of implication, which is the one that became
standard, would be introduced in print only in Heyting 1934 (p. 14);
in his paper 1932C, Heyting used the explanation given in Kolmogorov
1932 instead (see below).

Neither of these two papers by Heyting contained an argument for
the validity of Ex Falso.

A number of influences (or possible influences) on Heyting's
arriving at the Proof Interpretation can be suggested. The following
are publications Heyting had all seen by 1927, for he refers to them
in his dissertation (1925, 93–94):

Brouwer 1907 (Ch. 3) and Brouwer 1908, which forcefully made the
point that intuitionistic logic is concerned with the preservation of
constructibility;

Brouwer's proof of the bar theorem from 1924, handed in March 29,
1924 (Brouwer 1924D1, 189)/(Mancosu 1998, 36), and perhaps also the
later version from 1927, of which the manuscript was handed in on
April 28, 1926 (Brouwer 1927B, 75)/(van Heijenoort 1967, 446); both show
how to operate mathematically on demonstrations as objects.

Weyl 1921, where universal and existential theorems are considered
to be not genuine judgements at all, but
“Urteilsanweisungen” (judgement instructions) and
“Urteilsabstrakte” (judgement abstracts), thus emphasizing
that such theorems for their justification need to be backed up by a
construction method. Brouwer, in a note on Weyl's paper, agreed,
saying “This is only a matter of name and certainly does not
reflect any lacking insight on my part” (Mancosu 1998, 122).

A further likely influence is Brouwer's unpublished elucidation of the
virtual ordering axioms (see
section 3.1.3
above). Dirk van Dalen (personal communication) suspects that,
although Heyting was probably not present at this lecture course, he
heard Brouwer make a similar comment on another occasion. (An example
of such a possible occasion would be the period when Heyting was
working on his dissertation under Brouwer, for that work also
considers intuitionistic orderings.)

In 1932, Kolmogorov presented a logic of problems and their
solutions, and pointed out that the logic this explanation validates
is formally equivalent to the intuitionistic propositional and
predicate logic presented by Heyting in 1930. Moreover, he suggests
that this provides a better interpretation than Heyting's.

Kolmogorov's idea is this:

If a and b are two problems, then
a ∧ b designates the problem “to solve
both problems a and b”, while a ∨
b designates the problem “to solve at least one of the
problems a and b”. Furthermore, a
⊃ b is the problem “to solve b provided
that the solution for a is given” or, equivalently,
“to reduce the solution of b to the solution of
a” … ¬a designates the problem
“to obtain a contradiction provided that the solution of
a is given” … (x)a(x)
stands in general for the problem “to give a general method for
the solution of a(x) for every single value of
x”. (Kolmogorov 1932, 59)/(Mancosu 1998, 329)

He then lists Heyting's axioms for propositional logic (with Heyting's
numbering) and, by discussing an example, makes it clear that these
all hold when interpreted as statements about problems and their
solutions. He also points out that a ∨ ¬a is
the problem

to give a general method that allows, for every problem
a, either to find a solution for a, or to infer a
contradiction from the existence of a solution for a!

In particular, if the problem a consists in the proof of a
proposition, then one must possess a general method either to prove
or to reduce to a contradiction any proposition.
(Kolmogorov 1932, 63)/(Mancosu 1998, 332).

In the second part of his paper, Kolmogorov argues that, given
the epistemological tenets of intuitionism, “intuitionistic logic
should be replaced by the calculus of problems, for its objects are
in reality not theoretical propositions but rather problems”
(Kolmogorov 1932, 58)/(Mancosu 1998, 328). That he considers his
interpretation an alternative to Heyting's, and a preferable one, is
again emphasized in a note added in proof:

This interpretation of
intuitionistic logic is closely connected with the ideas Mr. Heyting
has developed in the last volume of Erkenntnis 2, 1931, 106
[Heyting 1931]; yet in Heyting a clear distinction between
propositions and problems is missing. (Kolmogorov 1932, 65)/(Mancosu
1998, 334)

But it is not at all clear that Heyting would want to make that
distinction. If the notion of proposition is understood in such a
way that a proposition is true or false independently of our
knowledge of this fact, then Heyting would readily agree with
Kolmogorov that a proposition is different from a problem; but as
soon as one adopts the view that propositions express intentions
that are fulfilled (i.e., made true) or disappointed (made false) by
our mathematical constructions, which is the view that Heyting
actually held, then there would seem to be no essential difference
between propositions and problems. Kolmogorov himself had already
indicated that a problem may consist in finding the proof of a
proposition; exploiting this, one can argue that the following two
notions of proposition coincide:

propositions express intentions towards constructions

propositions pose problems which are solved by carrying out constructions

The basic idea is that a proposition in sense 1 gives rise to the
problem of finding a construction that fulfills the expressed
intention, and that a solution to a problem posed in a proposition
in sense 2 also serves to fulfill the intention towards
constructions that solve that problem; this is made fully explicit
in a little argument due to Martin-Löf, given in detail in
Sundholm 1983 (pp. 158–159).

In a letter to Heyting of October 12, 1931, Kolmogorov in effect
agrees that the difference between Heyting and him is mainly a
terminological matter (Troelstra 1990, 15).

Heyting later claimed that Kolmogorov's meaning explanation and
his own amounted to the same (Heyting 1958C, 107). By 1937, Kolmogorov
seems to have come to believe the same, as in a review in
the Zentralblatt of an exchange between Freudenthal and
Heyting (discussed in a future update of this entry), he consistently
speaks of “intention or problem” (Kolmogorov 1937). In
that exchange itself, Freudenthal (1936, 119) had said that between
Heyting's and Kolmogorov's explanations there was “no essential
difference”. Finally, Oskar Becker, in a letter to Heyting of
September 1934, had remarked that Heyting's interpretation is a
generalization of Kolmogorov's, as a “problem” and its
“solution” are special cases of an intention and its
fulfillment. “Intuitionistic logic is therefore a
‘calculus of
intentions’”.[34]

However, a complication for the identification of Heyting's and
Kolmogorov's explanations of logic is introduced by Kolmogorov's also
accepting, in a particular case, solutions that do not consist in a
carrying out a concrete construction. Kolmogorov said that “As
soon as ¬a is solved, then the solution of a is
impossible and the problem a → b is without
content” (Kolmogorov 1932, 62)/(Mancosu 1998, 331), and proposed
that “The proof that a problem is without content [owing to an
impossible assumption] will always be considered as its
solution” (Kolmogorov 1932, 59)/(Mancosu 1998, 329). Taken
together, this yields a justification of Ex Falso, ¬a
→ (a → b).

It seems not altogether unreasonable to extend the meaning of the
term “solution” this way, for, just like a concrete
solution, an impossibility proof also provides what might be called
“epistemic closure”: like a concrete solution, it provides
a completely convincing reason to stop working on a certain
problem. (This kind of “higher-order” solution is also
familiar from Hilbert's Program, e.g., Hilbert 1900b (p. 51).) Note
that this justification of Ex Falso makes no attempt to describe a
counterfactual mathematical construction process; thus, Kolmogorov's
justification from 1932 is not incompatible with the ground for his
rejection of Ex Falso in 1925, namely, that one cannot constructively
assert consequences of something impossible. Rather, the solution from
1932 introduces a stipulation to achieve completion of the logical
theory for its own sake.

On the other hand, although Kolmogorov's stipulation is neither
unreasonable nor unmotivated, on Brouwer's descriptive conception of
logic there is of course no place for stipulation. For this reason,
“Proof Interpretation” seems to be a more appropriate name for an
explanation of Brouwerian logic than “BHK Interpretation”.

On Heyting's explanation, however, a justification of Ex Falso
parallel to Kolmogorov's would seem to be impossible: while a problem
may find a “higher-order” solution when it is shown that a
solution is impossible, it makes no sense to say that an intention
finds “higher-order” fulfillment when it is shown that it
cannot be fulfilled. The notion of a solution seems to permit a
reasonable extension that the notion of fulfillment does not. In his
book from 1934, Heyting explains Ex Falso in Kolmogorov's terms, not
his own. After stating the axiom ¬a ⊃ (a
⊃ b), he says:

It is appropriate to interpret the notion of
“reducing” in such a way, that the proof of the
impossibility of solving
a at the same time reduces the solution of any problem
whatsoever to that of a. (Heyting 1934, 15) [translation mine]

Clearly there is a difference between Kolmogorov's own explanation and
Heyting's explanation in Kolmogorov's terms. Where Heyting says that a
proof of ¬a establishes a reduction of the solution of
any problem to that of a, Kolmogorov had said that it
established that the problem of reducing the solution of any problem
to that of
a has become without content. One has the impression that Heyting
in his explanation of Ex Falso tries to approximate as closely as
possible the explanation for ordinary implications in terms of a
concrete constructive connection between antecedent and consequent;
this is even clearer in the explanation he would give of Ex Falso in
1956 (see
section 5.4 below).
(Note that neither Heyting nor Kolmogorov ever justified Ex Falso by
giving the traditional argument (based on the disjunctive syllogism)
also stated in Glivenko's paper from 1929 (see
section 4.3 above).)

More generally, the explanation of logic in Heyting 1934 is for
the most part given Kolmogorov style, and not Heyting's own in terms
of intentions and their fulfillment. (The latter is only mentioned
for its explanation of the implication (Heyting 1934, 14).) Perhaps
the reason for this is that Heyting (1934, 14) agrees with Kolmogorov
(1932, 58) that the interpretation in terms of problems and solutions
provides a useful interpretation Heyting's formal system also for
non-intuitionists (while for intuitionists they come to the same
thing). In his short note 1932C, titled “The application of
intuitionistic logic to the definition of completeness of a logical
calculus”, Heyting uses Kolmogorov's interpretation instead of his
own. Given the subject matter, that is what one might expect.

In his influential book Intuitionism. An introduction from
1956, Heyting explains the logical connectives as follows
(97–98, 102):

“A mathematical proposition p always demands a
mathematical construction with certain given properties; it can be
asserted as soon as such a construction has been carried
out.”

“p ∧ q can be asserted if and only if
both p and q can be asserted.”

“p ∨ q can be asserted if and only if
at least one of the propositions p and q can be
asserted.

“¬p can be asserted if and only if we possess a
construction which from the supposition that a construction p
were carried out, leads to a contradiction.”

“The implication p → q can be
asserted, if and only if we possess a construction r, which,
joined to any construction proving p (supposing that the
latter be effected), would automatically effect a construction
proving q.”

“⊢
(∀x)p(x) means
that p(x) is true for every x in Q
[over which x ranges]; in other words, we possess a general
method of construction which, if any element a of Q
is chosen, yields by specialization the
construction p(a).”

“(∃x)p(x) will be true if
and only if an element a of Q for
which p(a) is true has actually been
constructed.”

Note that these explanations are not in terms of proof conditions, but
of assertion conditions. This may make a difference in particular for
the explanation of implication, where, instead of only the information
under what condition something counts as a proof of p, we now
can also take into consideration that, by hypothesis, a concrete
construction for p has been effected. As we saw in
section 3.1.2, the
possibility to do so is crucial for Brouwer's proof of the Bar
Theorem.

In the same pages, Heyting also gave the following justification of
Ex Falso:

Axiom X [¬p → (p
→ q)] may not seem intuitively clear. As a matter of
fact, it adds to the precision of the definition of implication. You
remember that p → q can be asserted if and only
if we possess a construction which, joined to the
construction p, would prove q. Now suppose that
⊢ ¬p,
that is, we have deduced a contradiction from the supposition
that p were carried out. Then, in a sense, this can be
considered as a construction, which, joined to a proof of p
(which cannot exist) leads to a proof of
q. (Heyting 1956, 102)

One easily recognizes Heyting's effort to explain Ex Falso as much
as possible along the same lines as other implications, namely, by
providing a concrete construction that leads from the antecedent to
the consequent. In its attempt to provide, “in a sense”, a
construction, the explanation is clearly not of the same kind as
Kolmogorov's stipulation from 1932. But it does not fit into Heyting's
original interpretation of logic in terms of intentions directed at
constructions and the fulfillment of such intentions either. For to
fulfill an intention directed toward a particular construction we will
have to exhibit that construction; we will have to exhibit a
construction that transforms any proof of p into one of
q. But how can a construction that from the assumption
p arrives at a contradiction, and therefore generally
speaking not at q, lead to q? It will not do to say
that such a construction exists “in a sense”. A
construction that is a construction “in a sense”, as
Heyting helps himself to here, is no construction.

Brouwer's writings are referred to according to the scheme in the
bibliography van Dalen 1997a; Gödel's, according to the
bibliography in Gödel 1986, Gödel 1990, Gödel 1995
(except for Gödel 1970); Heyting's, according to the bibliography
Troelstra et al. 1981 (except for Heyting 1928).

van Atten, M., in press, “The hypothetical judgement in the
history of intuitionistic logic”, forthcoming in Logic, Methodology, and
Philosophy of Science XIII: Proceedings of the 2007 International
Congress in Beijing, C. Glymour, W. Wang, and D. Westerståhl,
eds., London: King's College Publications.

Acknowledgments

I am grateful to Dirk van Dalen and Göran Sundholm for
discussions of some of the issues involved. Helpful comments by the
editors and Rosalie Iemhoff led to various improved formulations and
clarifications. Dirk van Dalen kindly granted permission to quote from
materials in the Brouwer Archive. Thanks also to Dirk van Dalen and
Eckhart Menzler-Trott for their search for Bernays' letter to
Brouwer. In
section 3.6
I have drawn on van Atten 2004a; I thank the Association for Symbolic
Logic for granting permission to do this.

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